American Institute of Mathematical Sciences

Advanced
February  2019, 24(2): 831-849. doi: 10.3934/dcdsb.2018209

Boundedness in a three-dimensional Keller-Segel-Stokes system involving tensor-valued sensitivity with saturation

Dan Li 1,, , Chunlai Mu 1, , Pan Zheng 2, and  Ke Lin 3,

1. 

College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China
2. 

Department of Applied Mathematics, Chongqing University of Posts and Telecommunications, Chongqing 400065, China
3. 

College of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China

* Corresponding author: liwadanj@126.com

Received  October 2017 Revised  March 2018 Published  June 2018

Fund Project: The first author is partially supported by graduate research and innovation foundation of Chongqing, China (Grant No. CYB 17040). The second author is partially supported by NSFC (Grant Nos. 11771062, 11571062), the Basic and Advanced Research Project of CQC-STC (Grant No. cstc2015jcyjBX0007) and Fundamental Research Funds for the Central Universities (Grant Nos. 10611CDJXZ238826). The third author is partially supported by National Science Foundation of China (Grant Nos. 11601053, 11526042). The fourth author is partially supported by the Fundamental Research Funds for the Central Universities (Grant No. JBK1801059)

This paper deals with a boundary-value problem for a coupled chemotaxis-Stokes system with logistic source
$\begin{eqnarray*}\left\{\begin{array}{llll}n_t+u·\nabla n = \nabla·(D(n)\nabla n)-\nabla·(n \mathcal{S}(x, n, c)·\nabla c)\\ +ξ n-μ n^{2}, &x∈ Ω, &t>0, \\c_{t}+u·\nabla c = Δ c-c+n, &x∈Ω, &t>0, \\u_{t}+\nabla P = Δ u+n\nablaφ, &x∈Ω, &t>0, \\\nabla· u = 0, &x∈Ω, &t>0\end{array}\right.\end{eqnarray*}$
in three-dimensional smoothly bounded domains, where the parameters $ξ\ge0$, $μ>0$ and $φ∈ W^{1, ∞}(Ω)$, $D$ is a given function satisfying $D(n)\ge C_{D}n^{m-1}$ for all $n>0$ with $m>0$ and $C_{D}>0$. $\mathcal{S}$ is a given function with values in $\mathbb{R}^{3×3}$ which fulfills
$ \begin{equation*}{\label{1.3}}\begin{split}|\mathcal{S}(x, n, c)|\leq C_{\mathcal{S}}(1+n)^{-α}\end{split}\end{equation*}$
with some $C_{\mathcal{S}}>0$ and $α>0$. It is proved that for all reasonably regular initial data, global weak solutions exist whenever $m+2α>\frac{6}{5}$. This extends a recent result by Liu el at. (J. Diff. Eqns, 261 (2016) 967-999) which asserts global existence of weak solutions under the constraints $m+α>\frac{6}{5}$ and $m\ge\frac{1}{3}$.
Keywords: Chemotaxis- Stokes, nonlinear diffusion, tensor-valued sensitivity, boundedness, logistic source.
Mathematics Subject Classification: Primary: 35K45; Secondary: 35B65, 35Q92, 35Q35, 92C17.
Citation: Dan Li, Chunlai Mu, Pan Zheng, Ke Lin. Boundedness in a three-dimensional Keller-Segel-Stokes system involving tensor-valued sensitivity with saturation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 831-849. doi: 10.3934/dcdsb.2018209
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References:
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K. Baghaei and M. Hesaaraki, Global existence and boundedness of classical solutions for a chemotaxis model with logstic source, C. R. Acad. Sci. Paris. Ser. I, 351 (2013), 585-591. doi: 10.1016/j.crma.2013.07.027. Google Scholar

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[10]

M. Herrero and J. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Sc. Norm. Super. Pisa Cl. Sci., 24 (1997), 633-683. Google Scholar

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[13]

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[14]

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[15]

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[17]

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[18]

S. Ishida and T. Yokota, Blow-up finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type, Discrete Continuous Dynam. Systems - B, 18 (2013), 2569-2596. doi: 10.3934/dcdsb.2013.18.2569. Google Scholar

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E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5. Google Scholar

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A. Kiselev and L. Ryzhik, Biomixing by chemotaxis and efficiency of biological reactions: the critical reaction case, J. Math. Phys., 53 (2012), 115609, 9pp. doi: 10.1063/1.4742858. Google Scholar

[21]

A. Kiselev and L. Ryzhik, Biomixing by chemotaxis and enhancement of biological reactions, Comm. Partial Differential Equations, 37 (2012), 298-318. doi: 10.1080/03605302.2011.589879. Google Scholar

[22]

R. Kowalczyk and Z. Szymańska, On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343 (2008), 379-398. doi: 10.1016/j.jmaa.2008.01.005. Google Scholar

[23]

H. A. Levine and B. D. Sleeman, A system of reaction-diffusion equations arising in the theory of reinforced randomw walks, SIAM J. Appl. Math., 57 (1997), 683-730. doi: 10.1137/S0036139995291106. Google Scholar

[24]

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[25]

X. Li, Y. Wang and Z. Xiang, Global existence and boundedness in a 2D KellerSegel-Stokes system with nonlinear diffusion and rotational flux, Commun. Math. Sci., 14 (2016), 1889-1910. doi: 10.4310/CMS.2016.v14.n7.a5. Google Scholar

[26]

X. Li and Y. Xiao, Global existence and boundedness in a 2D Keller-Segel-Stokes system, Nonlinear Anal., 37 (2017), 14-30. doi: 10.1016/j.nonrwa.2017.02.005. Google Scholar

[27]

J. Liu and Y. Wang, Global existence and boundedness in a Keller-Segel- (Navier-)Stokes system with signal-dependent sensitivity, J. Math. Anal. Appl., 447 (2017), 499-528. doi: 10.1016/j.jmaa.2016.10.028. Google Scholar

[28]

J. Liu and Y. Wang, Boundedness and decay property in a three-dimensionlal Keller-Segel-Stokes system involving tensor-valued sensitivity with saturation, J. Differential Equations, 261 (2016), 967-999. doi: 10.1016/j.jde.2016.03.030. Google Scholar

[29]

T. Nagai, Blow-up of nonradial solutions to parabolic-elliptic systems modelling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55. Google Scholar

[30]

K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal., 51 (2002), 119-144. doi: 10.1016/S0362-546X(01)00815-X. Google Scholar

[31]

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